Just to ask, if I have a colorless commander I can use any type of mana , but I will only produce colorless mana right.

Yes, but you can't include most lands. You need colorless mana producers because anything that produces black mana has a color identity of black, same with each other color. This is part of the reason colorless decks see almost no play, because the mana base is such a pain.

However, anything that says "produce mana of any color", or anything that doesn't specify any colors (City of Brass and Exotic Orchid, for example) is fair game.

Sure the mana base can be somewhat tricky but there is a lot of cheap lands with somewhat good abilities that don't come into play tapped and can give you colorless mana. The biggest reason to stay away from a colorless deck is in my opinion because of the commanders. They are all a bit too strong and threatening. Ulamog, the Infinite Gyre, Kozilek, Butcher of Truth & Karn, Silver Golem are all famous to sooner or later eat someone's lands.